Answer :
To find the absolute maximum and minimum values of the function [tex]\( f(x) = 9x^3 - 54x^2 + 81x + 13 \)[/tex] on the interval [tex]\([-6, 2]\)[/tex], let's go through the following steps:
1. Find the Critical Points:
- First, find the derivative [tex]\( f'(x) \)[/tex]. The derivative of the function is calculated as follows:
[tex]\[
f'(x) = 27x^2 - 108x + 81
\][/tex]
- Set the derivative equal to zero to find the critical points:
[tex]\[
27x^2 - 108x + 81 = 0
\][/tex]
- Simplifying the equation will help you find the values of [tex]\( x \)[/tex].
- The roots of this quadratic equation are the critical points.
2. Evaluate Critical Points and Endpoints:
- Check which critical points lie within the interval [tex]\([-6, 2]\)[/tex].
- Evaluate [tex]\( f(x) \)[/tex] at each critical point within the interval.
- Evaluate [tex]\( f(x) \)[/tex] at the endpoints [tex]\( x = -6 \)[/tex] and [tex]\( x = 2 \)[/tex].
3. Compare Values:
- After evaluation, compare all these values to determine which is the largest (maximum) and smallest (minimum).
- The one with the highest value is the absolute maximum, and the one with the lowest value is the absolute minimum.
After evaluating these values, we find:
- The function has an absolute maximum at [tex]\( x = 1 \)[/tex] where [tex]\( f(1) = 49 \)[/tex].
- The function has an absolute minimum at [tex]\( x = -6 \)[/tex] where [tex]\( f(-6) = -4361 \)[/tex].
Thus, the correct answer is:
D. [tex]\(\max f(x) = f(1) = 49\)[/tex]
[tex]\(\min f(x) = f(-6) = -4361\)[/tex]
1. Find the Critical Points:
- First, find the derivative [tex]\( f'(x) \)[/tex]. The derivative of the function is calculated as follows:
[tex]\[
f'(x) = 27x^2 - 108x + 81
\][/tex]
- Set the derivative equal to zero to find the critical points:
[tex]\[
27x^2 - 108x + 81 = 0
\][/tex]
- Simplifying the equation will help you find the values of [tex]\( x \)[/tex].
- The roots of this quadratic equation are the critical points.
2. Evaluate Critical Points and Endpoints:
- Check which critical points lie within the interval [tex]\([-6, 2]\)[/tex].
- Evaluate [tex]\( f(x) \)[/tex] at each critical point within the interval.
- Evaluate [tex]\( f(x) \)[/tex] at the endpoints [tex]\( x = -6 \)[/tex] and [tex]\( x = 2 \)[/tex].
3. Compare Values:
- After evaluation, compare all these values to determine which is the largest (maximum) and smallest (minimum).
- The one with the highest value is the absolute maximum, and the one with the lowest value is the absolute minimum.
After evaluating these values, we find:
- The function has an absolute maximum at [tex]\( x = 1 \)[/tex] where [tex]\( f(1) = 49 \)[/tex].
- The function has an absolute minimum at [tex]\( x = -6 \)[/tex] where [tex]\( f(-6) = -4361 \)[/tex].
Thus, the correct answer is:
D. [tex]\(\max f(x) = f(1) = 49\)[/tex]
[tex]\(\min f(x) = f(-6) = -4361\)[/tex]
